medical imaging instrumentation & image analysis
DESCRIPTION
Medical Imaging Instrumentation & Image Analysis. Atam P. Dhawan, Ph.D. Dept. of Electrical & Computer Engineering Dept. of Biomedical Engineering New Jersey Institute of Technology Newark, NJ, 07102 [email protected]. Imaging in Medical Sciences. - PowerPoint PPT PresentationTRANSCRIPT
Medical Imaging Instrumentation & Image Analysis
Atam P. Dhawan, Ph.D.
Dept. of Electrical & Computer EngineeringDept. of Biomedical Engineering
New Jersey Institute of TechnologyNewark, NJ, 07102
Imaging in Medical Sciences
Imaging is an essential aspect of medical sciences for visualization of anatomical structures and functional or metabolic information of the human body.
Structural and functional imaging of human body is important for understanding the human body anatomy, physiological processes, function of organs, and behavior of whole or a part of organ under the influence of abnormal physiological conditions or a disease.
Medical Imaging Radiological sciences in the last two decades have
witnessed a revolutionary progress in medical imaging and computerized medical image processing.
Advances in multi-dimensional medical imaging modalities X-ray Mammography X-ray Computed Tomography (CT) Single Photon Computed Tomography (SPECT) Positron Emission Tomography (PET) Ultrasound Magnetic Resonance Imaging (MRI) functional Magnetic Resonance Imaging (fMRI)
Important radiological tools in diagnosis and treatment evaluation and intervention of critical diseases for significant improvement in health care.
A Multidisciplinary Paradigm
Physiology and CurrentUnderstanding
Physics of Imaging
Instrumentationand Image Acquisition
Computer Processing,Analysis and Modeling
Applications andIntervention
Electromagnetic Radiation Spectrum
10-10
Radio Waves
TV Waves
Radar Waves
Microwaves Infrared Rays
Visible Light
Ultraviolet Rays
X-rays Gamma Rays
102 101 1 10-1 10-2 10-3 10-4 10-6 10-7 10-8
Wavelength in meters
Frequency in Hz
10-5 10-9 10-10 10-11 10-12 10-13 10-14 103
106 107 109 1010 1011 1012 1014 1015 1016 1013 1017 1018 1019 1020 1021 1022 105 108
Energy in eV
10-9 10-8 10-6 10-5 10-4 10-3 10-1 1 101
10-2 102 103 104 105 106 107 10-7
MRI
X-ray Imaging
Gamma-ray Imaging
Cosmic Rays
Medical Imaging Information Anatomical
X-Ray Radiography X-Ray CT MRI Ultrasound Optical
Functional/Metabolic SPECT PET fMRI, pMRI Ultrasound Optical Fluorescence Electrical Impedance
Medical Imaging ModalitiesMedical Imaging Methods
Internal Combination:
Internal
& External
External
UsingEnergy Source
SPECTPET
MRIFluorescenceEI
X-RayUltrasoundOptical
Medical Imaging Thru Transmission
X - r ay
R ad iogr aphs
X - r ay
Comput ed
T omogr aphy
I nf r ar ed O pt ic al M ic r owave
T r ansmiss ion
of E lec t r o-
magnet ic
E ner gy
Basic Principle: Radiation is attenuated when passed through the body.
ROC: Performance Measure
Ntp = Notp + Nofn and Ntn = Nofp + Notn
FractionsTrue Positive Fraction (TPF): ratio of the number of positive observations to the
number of positive true-condition cases.
TPF = Notp/Ntp
False Negative Fraction (FNF): ratio of the number of negative observations to the number of positive true-condition cases.
FNF = Nofn/Ntp\
False Positive Fraction (FPF): ratio of the number of positive observations to the number of negative true-condition cases.
FPF = Nofp/Ntn
True Negative Fraction (TNF): ratio of the number of negative observations to the number of negative true-condition cases.
TNF = Notn/Ntn
TPF + FNF = 1 and TNF + FPF = 1
Measures
Sensitivity is TPF
Specificity is TNF
Accuracy = (TPF+TNF)/Ntotal
ROC Curve
FPF=1-TNF
TPF
b
a
c
A Case StudyTotal number of patients = Ntot=100
Total number of patients with biopsy proven cancer (true condition of object present) = Ntp=10
Total number of patients with biopsy proven normal tissue (true condition of object NOT present) = Ntn=90
Out of the patients with cancer Ntp , the number of patients diagnosed by the physician as having cancer = Number of True Positive cases = Notp=8
Out of the patients with cancer Ntp, the number of patients diagnosed by the physician as normal = Number of False Negative cases = Nofn=2
Out of the normal patients Ntn, the number of patients rated by the physician as normal = Number of True Negative cases = Notn=85
Out of the normal patients Ntn, the number of patients rated by the physician as having cancer = Number of False Positive cases = Nofp=5
Example
True Positive Fraction (TPF) = 8/10 = 0.8
False Negative Fraction (FNF) = 2/10 = 0.2
False Positive Fraction (FPF) = 5/90 = 0.0556
True Negative Fraction (TNF) = 85/90 = 0.9444
Linear System
A system is said to be linear if it follows two properties: scaling and superposition.
)},,({)},,({)},,(),,({ 2121 zyxIbhzyxIahzyxbIzyxaIh
Image Formation: Object f, Image g
0),,(
0),,(
zyxg
f
g1(x, y,z) + g2(x, y,z) = h(x,y,z, , f1 ()) + h(x,y,z, , f2 ())
h(x,y,z, , f ()) = h(x,y,z, ) f ()
Non-negativity
Superposition
Linear Response Function
dddfzyxhzyxg )),,(,,,,,,(),,(
Image Formation
dddfzyxhzyxg ),,(),,,,,(),,(
Linear Image Formation
Image Formation
g z
Image Formation System
hObject Domain
Image Domain
a
b
x
y
Radiating Object f( , ,a b g) Image g(x,y,z)
dddfzyxhzyxg ),,(),,,,,(),,(
dddfzyxhzyxg ),,(),,(),,(
Simple Case: LSI Image Formation
ddfyxhyxg ),(),(),(
g=h**f
Image Formation: External Source
Reconstructed Cross-Sectional Image
Radiation Source
gz
Image Formation
Systemh
Object Domain
Image Domain
Selected Cross-Section
b
x
yRadiating Object
a
Image
Image Formation: Internal Source
a
gz
Reconstructed Cross-Sectional Image
Image Formation
Systemh
Object Domain
Image Domain
Selected Cross-Section
b
x
y
Radiating Object
Image
Fourier Transform
dydxeyxgyxgFTvuG vyuxj ,),()},({),( )(2
dudvevuGvuGFTyxg vyuxj
)(21 ),()},({),(
Properties of FT
b
v
a
uG
abbyaxg ,
1)},(FT
Scaling: It provides a proportional scaling.
FT {ag(x,y)+bh(x,y)}= aFT{g(x,y)+ bFT{h(x,y)
Linearity: Fourier transform, FT, is a linear transform.
FT Properties
)(2),()},({ vbuajevuGbyaxgFT
),(),(),(),( vuHvuGddyxhgFT
),(*),(),(*),( vuHvuGddyxhgFT
Translation
Convolution
Cross-Correlation
FT Properties….
2),(),(*),(),(*),( vuGvuGvuGddyxggFT
dudvvuGvuGdxdyyxgyxg ),(*),(),(*),(
)}({)}({)},({
)()(),(
ygFTxgFTyxgFT
then
ygxgyxg
yyxx
yx
Auto-Correlation
Parseval’s Theorem
Separability
Radon Transform
x
y
q
q
p
q
p
f(x,y)
P(p,q)
Line integral projection P(p,q) of the two-dimensional Radon transform.
Radon TransformProjection p1
Projection p2
Projection p3
Reconstruction Space
A
B
dqqpqpfpJyxfR )cossin,sincos()()},({
Backprojection Reconstruction Method
L
i
pJL
yxf
pdpphpJpJ
i1
*
)(),(
)()()(
where L is the total number of projections acquired during the imaging process at viewing angles
L .., 1, ifor ; i
Sampling Theorem
The sampling theorem provides the mathematical foundation of Nyquist criterion to determine the optimal sampling rate for discretization of an analog signal without the loss of any frequency information.
The Nyquist criterion states that to avoid any loss of information or aliasing artifact, an analog signal must be sampled with a sampling frequency that is at least twice the maximum frequency present in the original signal.
Sampling
1 2
),(),( 21j j
yjyxjxyxs
1 2
),(),(),(),(f y][x,f 2121adj j
a yjyxjxyjxjfyxsyx
The sampled version of the image, fd[x,y] is obtained from sampling the analog version as
Sampling Effect In Fourier domain the spectrum overlapping has to be
avoided by proper sampling of the image in spatial domain. Sampling in spatial domain produces a convolution in the
frequency domain.
xs = 2/x and xy = 2/y
1 2
),(1
),( 21j j
ysyxsxayxs jjFyx
F
where Fa(x,y) is the Fourier transform of the analog image fa(x,y) and xs and xy represent the Fourier domain sampling spatial frequencies such that
To avoid overlapping of image spectra, it is necessary that
xs >= xmax = 2fxmax and ys >= ymax = 2fymax
Nyquist (Optimal) Sampling
wx
wy
wymax
wxmax-wxmax
-wymax
Fa(wx,wy)
(a)
(b)
(c)
xs >= xmax = 2fxmax and ys >= ymax = 2fymax
Wavelet Transform
Fourier Transform only provides frequency information.
Windowed Fourier Transform can provide time-frequency localization limited by the window size.
Wavelet Transform is a method for complete time-frequency localization for signal analysis and characterization.
Wavelet Transform..
Wavelet Transform : works like a microscope focusing on finer time resolution as the scale becomes small to see how the impulse gets better localized at higher frequency permitting a local characterization
Provides Orthonormal bases while STFT does not.
Provides a multi-resolution signal analysis approach.
Wavelet Transform…
Using scales and shifts of a prototype wavelet, a linear expansion of a signal is obtained.
Lower frequencies, where the bandwidth is narrow (corresponding to a longer basis function) are sampled with a large time step.
Higher frequencies corresponding to a short basis function are sampled with a smaller time step.
Continuous Wavelet Transform
Shifting and scaling of a prototype wavelet function can provide both time and frequency localization.
Let us define a real bandpass filter with impulse response y(t) and zero mean:
This function now has changing time-frequency tiles because of scaling. a<1: y(a,b) will be short and of high frequency a>1: y(a,b) will be long and of low frequency
a
bt
at
tft
dttfa
bt
abaCWT
dtt
ba
ba
R
f
1)( where
)(),(
)(*1
),(
as defined is (CWT) Transform Wavelet contnuousA
0)0()(
,
,
Wavelet Decomposition
T h e w a v e l e t t r a n s f o r m o f a s i g n a l i s i t s d e c o m p o s i t i o n o n a f a m i l y o fr e a l o r t h o n o r m a l b a s e s
m n ( x ) o b t a i n e d t h r o u g h t r a n s l a t i o n a n dd i l a t i o n o f a k e r n e l f u n c t i o n ( x ) k n o w n a s t h e m o t h e r w a v e l e t .
W h e r e m , n Z , a s e t o f i n t e g e r s
)2(2)( 2/, nxx mm
nm
Wavelet Coefficients
Using orthonormal property of the basis functions, wavelet coefficients of a signal f(x) can be computed as
The signal can be reconstructed from the coefficients as
)()()( ,, xdxxfd nmnm
)()( ,, xdxf nmm n
nm
Wavelet Transform with Filters The mother wavelet can be constructed using a scaling
function f(x) which satisfies the two-scale equation
Coefficients h(k) have to meet several conditions for the set of basis functions to be unique, orthonormal and have a certain degree of regularity.
For filtering operations, h(k) and g(k) coefficients can be used as the impulse responses correspond to the low and high pass operations.
)()1()(
)2()(2)(
)2()(2)(
klk
where
nxnx
nxnx
k
n
n
hg
g
h
Decomposition
H H
G
H
G
G
2 2
2
2
2
Data
Wavelet Decomposition Space
V0 data
V1 W1
V2 W2
V3 W3
Image Decomposition
h g
sub-sample
Level 0 Level 1
h- h
h-g
g-h
g-g
horizontally vertically
sub-sample
g
gh
h
XImage
Wavelet and Scaling Functions
Image Processing and Enhancement